The Fractal Distribution of Galaxies and the Transition to Homogeneity

There is an ongoing debate in cosmology about the value of the length scaleat which ``homogeneity'' in the matter distribution is reached or even if sucha scale exists. In the wake of this debate, we intend in this letter to clarifythe meaning of the statement transition to homogeneity and of the concept ofcorrelation length. We show that there are two scales each associated to thefractal and to the homogeneous regimes of the matter distribution,respectively. The distinction between both scales has deep consequences, forthere can be a regime which despite having small fluctuations around theaverage density exhibits large clusters of galaxies.Comment: 10 pages, 2 large PS figures, AAS macros. Substantial additions as regards the implications for cosmology, including the prospective observation of critical phenomena, and new references. Version to appear in ApJ